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On random fixed point theorems with applications to integral equations
Heliyon 5 (2019) e01641
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On random fixed point theorems with applications to integral equations Kanayo Stella Eke *, Hudson Akewe, Sheila Amina Bishop Department of Mathematics, Covenant University, P. M. B. 1023, Ota, Ogun State, Nigeria
A R T I C L E I N F O
A B S T R A C T
Keyword: Applied mathematics
This particular research establishes some random fixed point theorems for general nonlinear random contractive operators in the context of partially ordered separable metric spaces. The existence and uniqueness of the random solution for the nonlinear integral equation is obtained by applying the result of the random fixed point. The results generalize and improve on some related works in the literature.
Our theorems are proved in the context of metric space while Saluja and Tripathi [8] proved in the context of partial metric spaces. Nieto, Ouahab and. Rodriguez-Lopez [9] proved their theorem using Banach contraction mappings while we proved our theorems using Hardy and Rogers contraction mappings. Rashwan and Albaqeri [3] proved the solution of the random integral equation using the Banach contraction operator while we proved our solution to the random integral equations employing more general contractive operator.
1. Introduction FIXED point theory plays very fundamental role in solving deterministic operator equations. An important principle that is required to solve applied mathematical problems in applied mathematics, such as computer sciences involve looping. Fixed point iteration and monotonous iteration techniques introduced by Banach [1] and Ran and Reurings [2] respectively handled these problems. Probabilistic fundamental analysis is an important aspect of mathematics that is applied to solving problems. An equation that requires a mathematical model to describe its phenomena is classified as random equation. Fixed point theorems for stochastic functions was pioneered by Prague School of Probability. In 1955–1965, Spacek and Hans studied the Fredholm integral equations with respect to random kernel. Stochastic fixed point theorem for contractive mappings in Polish spaces were established by Rashwan and Albaqeri [3], Hans [4], Hans and Spacek [5], Okeke and Eke [6]. In 1977, Lee and Padgett [7] proved fixed point result for stochastic nonlinear contractive mappings in separable Banach spaces and applied the result obtained to establish the existence and uniqueness of the random solution of the stochastic nonlinear integral equations. Saluja and Tripathi [8] proved some stochastic fixed point results for contractive mappings in the framework of cone random metric spaces.
Nieto et al. [9] established a stochastic fixed point theorem for Banach contraction mappings in ordered metric spaces. In the same reference, the result obtained is used to established the solution for random differential equations with boundary properties. This particular paper proves the existence and uniqueness of random fixed point for a more general nonlinear contractive functions in partially ordered separable metric spaces. The result obtain is apply to establish the existence and uniqueness of random solution for nonlinear integral equation. 2. Methodology Suppose ðA; ξÞ is a separable Banach space, where ξ is aσ -algebra of Borel subsets of A and ðϕ; ξ; μÞ represents a complete probability measure space with measureμ. ξ is also a σ -algebra of subsets of ϕ. The following definitions are found in Joshi and Bose [10] with more details for the readers. Definition 1.1. An operator T : ϕ A → B is known as a random operator if Tðv; aÞ ¼ BðvÞ is a random variable for every a 2 A. Definition 1.2. An operator T : ϕ A → B is called continuous random mapping if the set of all v 2 ϕ for which Tðv; aÞ is a continuous function of
* Corresponding author. E-mail address: [email protected] (K.S. Eke). https://doi.org/10.1016/j.heliyon.2019.e01641 Received 17 October 2018; Received in revised form 29 April 2019; Accepted 30 April 2019 2405-8440/© 2019 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article as: K.S. Eke et al., On random fixed point theorems with applications to integral equations, Heliyon, https://doi.org/10.1016/ j.heliyon.2019.e01641
K.S. Eke et al.
Heliyon 5 (2019) e01641
a has measure one.
Definition 1.3. An equation of the type Tðv; aðvÞÞ ¼ aðvÞ, where T : ϕ A → A is a random mapping is called a random fixed point equation.
mðvÞ þ nðvÞ þ qðvÞ dðbn1 ðvÞ; bn ðvÞÞ 1 nðvÞ qðvÞ
kdðbn1 ðvÞ; bn ðvÞÞ:
Definition 1.4. Any mapping a : ϕ → A which satisfies random fixed point equation Tðv; aðvÞÞ ¼ aðvÞ almost surely is called a wide sense solution of the fixed point equation.
Where k ¼
mðvÞþnðvÞþqðvÞ 1nðvÞqðvÞ
< 1:
Consequently, we get
Definition 1.5. Any A-valued random variable aðvÞ which satisfies μfv : Tðv; aðvÞÞ ¼ aðvÞg ¼ 1 is called random solution of the fixed point equation. Let A be a nonempty set and let T : A → A be an operator. We define the Picard iteration of T by T n ¼ TðT n1 Þ; for n 2 ℕ; n 2:
dðbn ðvÞ; bnþ1 ðvÞÞ ½kðvÞn dðb0 ðvÞ; b1 ðvÞÞ: For n > m we obtain, dðbm ðvÞ; bn ðvÞÞ dðbn ðvÞ; bmþ1 ðvÞÞ þ dðbmþ1 ðvÞ; bmþ2 ðvÞÞ þ ⋯ þ dðbn1 ðvÞ; bn ðvÞÞ
Definition 1.6. [9]: If ðA; Þis a partially ordered set and T : A → A , we say that T is monotone non decreasing if a b;a;b 2 A ⇒ TðaÞ TðbÞ.
ð½kðvÞm þ ½kðvÞ
3. Results
mþ1
This section proves the existence and uniqueness of random fixed point for contractive mappings in partially ordered separable metric spaces.
mþn1 þ ⋯ þ kðvÞ Þdðb0 ðvÞ; b1 ðvÞ
½kðvÞm dðb0 ðvÞ; b1 ðvÞÞ 1 kðvÞ
This shows that fbn ðvÞgn2ℕ is a Cauchy sequence for every v 2 ϕ. Let bðvÞ be the limit point of fbn ðvÞg, v 2 ϕ. Since b0 ðvÞ is measurable, then b1 ðvÞ is measurable. Consequently, for each n 2 ℕ, the operator v → bn ðvÞ is measurable. This implies that bðvÞ is measurable. Next we prove that bðvÞ is the random fixed point of T.
Theorem 3.1. Let ðϕ; TÞ be a measurable space and ðA; d; αÞ a complete partially ordered separable metric space. If T : ϕ A → A is a continuous random operator such that, Tðv; Þ is a monotonous operator. If the following conditions hold: (J1) For each v 2 ϕ, there exists mðvÞ þnðvÞ þ 2qðvÞ < 1 such that dðTðv; a1 Þ; Tðv; a2 ÞÞ mðvÞdða1 ; a2 Þ þnðvÞ½dða1 ; Tðv; a1 ÞÞ þ dða2 ; Tðv; a2 ÞÞ þqðvÞ½dða1 ; Tðv; a2 ÞÞ þ dða2 ; Tðv; a1 ÞÞ:
(2)
dðbðvÞ; Tðv; bðvÞÞ dðbðvÞ; bn ðvÞÞ þ dðbn ðvÞ; Tðv; bðvÞÞ dðbðvÞ; bn ðvÞÞ þ dðTðv; bn1 ðvÞÞ; Tðv; bðvÞÞÞ dðbðvÞ; bn ðvÞÞ þ mðvÞdðbn1 ðvÞ; bðvÞÞ þ nðvÞ½dðbn1 ðvÞ; Tðv; bn1 ðvÞÞ þ dðbðvÞ; Tðv; bðvÞÞ þ qðvÞ½dðbn1 ðvÞ; Tðv; bðvÞÞ þ dðbðvÞ; Tðv; bn1 ðvÞÞ:
(1)
As n → ∞, we get
For each a1 ; a2 2 A; a1 a2 : (J2) There is a random variable a0 : ϕ → A with
dðbðvÞ; Tðv; bðvÞÞ dðbðvÞ; ðvÞÞ þ mðvÞdðbðvÞ; bðvÞÞ þ nðvÞ½dðbðvÞ; Tðv; bðvÞÞ þ dðbðvÞ; Tðv; bðvÞÞ þ qðvÞ½dðbðvÞ; Tðv; bðvÞÞ þ dðbðvÞ; Tðv; bðvÞÞ
a0 ðvÞ Tðv; a0 ðvÞÞ; for all v 2 ϕ; or
ð2nðvÞ þ 2qðvÞÞdðbðvÞ; Tðv; bðvÞÞ
a0 ðvÞ Tðv; a0 ðvÞÞ; for all v 2 ϕ:
Since 2nðvÞ þ 2qðvÞ < 1 then it is a contradiction. Therefore bðvÞ is the random fixed point of T. To prove the uniqueness of the random fixed point of T, we consider the following proposition in [2]. (J3) Every pair of elements of A has a lower bound or upper bound. In [2], the above condition [J3] is equivalent to: [J3’] for every a; b 2 A, there is c 2 A that is comparable to a and b. The following theorem concludes that the existence of a random variable b : ϕ → A in Theorem 3.1 is the unique random fixed point of T.
Then there is a random variable a : ϕ → A which is a random fixed point of T. Proof: If for v 2 ϕ, there is a random variable a0 ðvÞ 2 ðϕ; TÞsuch that Tðv; a0 ðvÞÞ ¼ a0 ðvÞthen a0 ðvÞis a random fixed point of T. On the contrary, we assume that Tðv; a0 ðvÞÞ 6¼ a0 ðvÞ for some v 2 ϕ. Let b0 ðvÞ ¼ a0 ðvÞ, then we can define a sequence Tðv;bn1 ðvÞÞ ¼ bn ðvÞ, for v 2 ϕ, n 2 ℕ. According to (J1) we have, for each v 2 ϕ, and n 2 ℕ one of the following relations hold: bn ðvÞ bnþ1 ðvÞ or bn ðvÞ bnþ1 ðvÞ. Using the above inequality in Eq. (1) we obtain,
Theorem 3.2. Let all the hypotheses of Theorem 3.1 be satisfied and if the following condition [J3] (equivalently [J3’]) be satisfied, then b : ϕ → A is the unique random fixed point of T.
dðbn ðvÞ; bnþ1 ðvÞÞ dðTðv; bn1 ðvÞÞ; Tðv; bn ðvÞÞÞ mðvÞdðbn1 ðvÞ; bn ðvÞÞ þnðvÞ½dðbn1 ðvÞ; Tðv; bn1 ðvÞÞÞ þ dðbn ðvÞ; Tðv; bn ðvÞÞÞ þqðvÞ½dðbn1 ðvÞ; Tðv; bn ðvÞÞÞ þ dðbn ðvÞ; Tðv; bn1 ðvÞÞÞ
'
Proof: Let b : ϕ → A be an arbitrary random variable and we define the sequence ' ' ' ' b 0 ðvÞ ¼ a 0 ðvÞ; bn ðvÞ ¼ T v; b n1 ðvÞ ; v 2 ϕ; n 2 ℕ:
mðvÞdðbn1 ðvÞ; bn ðvÞÞ þnðvÞ½dðbn1 ðvÞ; bn ðvÞÞ þ dðbn ðvÞ; bnþ1 ðvÞÞ þqðvÞ½dðbn1 ðvÞ; bnþ1 ðvÞÞ þ dðbn ðvÞ; bn ðvÞÞ
In Theorem 3.1, we obtain that fbn ðvÞgn2ℕ → bðvÞ as n → ∞, for each '
v 2 ϕ, where bðvÞ is the random fixed point of T. If b 0 ðvÞ is comparable to
mðvÞdðbn1 ðvÞ; bn ðvÞÞ þnðvÞ½dðbn1 ðvÞ; bn ðvÞÞ þ dðbn ðvÞ; bnþ1 ðvÞÞ þqðvÞ½dðbn1 ðvÞ; bn ðvÞÞ þ dðbn ðvÞ; bnþ1 ðvÞÞ
b0 ðvÞ for each v 2 ϕ, it follows that Tðv; b0 ðvÞÞ' is comparable to Tðv; b0 ðvÞÞ '
for each v 2 ϕ. Therefore b n ðvÞ is comparable to bn ðvÞ for each v 2 ϕ. Hence,
ðmðvÞ þ nðvÞ þ qðvÞÞdðbn1 ðvÞ; bn ðvÞÞ þ ðnðvÞ þ qðvÞÞdðbn ðvÞ; bnþ1 ðvÞÞ
2
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' ' d bn ðvÞ;b n ðvÞ d Tðv;bn1 ðvÞÞ;T v;b n1 ðvÞ ' mðvÞd bn1 ðvÞ;b n1 ðvÞ
Similarly, we obtain
mðvÞ þ nðvÞ þ qðvÞ
' ' d bðvÞ; c n ðvÞ d bðvÞ; c n1 ðvÞ : 1 nðvÞ qðvÞ
þnðvÞ½dðb n1 ðvÞ;Tðv;b n1 ðvÞÞÞ ' ' þd b n1 ðvÞ;T v;b n1 ðvÞ ' þqðvÞ d bn1 ðvÞ;T v;b n1 ðvÞ ' þd b n1 ðvÞ;Tðv;bn1 ðvÞÞ
Choose a natural number N1 such that dðbðvÞ; cn ðvÞÞ
Therefore, ' ' d bðvÞ; b ðvÞ dðbðvÞ; cn1 ðvÞÞ þ d cn1 ðvÞ; b ðvÞ
As n → ∞, we get
mðvÞ þ nðvÞ þ qðvÞ εð1 nðvÞ qðvÞÞ 1 nðvÞ qðvÞ 2ðmðvÞ þ nðvÞ þ qðvÞÞ
mðvÞ þ nðvÞ þ qðvÞ εð1 nðvÞ qðvÞÞ þ 1 nðvÞ qðvÞ 2ðmðvÞ þ nðvÞ þ qðvÞÞ
d bðvÞ; bðvÞÞ mðvÞdðbðvÞ; b ðvÞ ' ' þ nðvÞ dðbðvÞ; bðvÞÞ þ dðb ðvÞ; b ðvÞÞ ' ' þ qðvÞ dðbðvÞ; b ðvÞÞ þ dðb ðvÞ; bðvÞÞ '
(3)
and
εð1 nðvÞ qðvÞÞ ' : d bðvÞ; c n ðvÞ 2ðmðvÞ þ nðvÞ þ qðvÞÞ
' ¼ mðvÞd bn1 ðvÞ; b n1 ðvÞ ' ' þnðvÞ dðbn1 ðvÞ; bn ðvÞÞ þ d b n1 ðvÞ; b n ðvÞ ' ' þqðvÞ d bn1 ðvÞ; b n ðvÞ þ d b n1 ðvÞ; bn ðvÞ :
εð1 nðvÞ qðvÞÞ 2ðmðvÞ þ nðvÞ þ qðvÞÞ
'
¼
ε 2
ε
þ ¼ ε: 2
Since ε is the smallest positive number we assumeε ¼ 0. Therefore '
'
'
dðbðvÞ; b ðvÞÞ ¼ 0. Thus bðvÞ ¼ b ðvÞ: The uniqueness proved.
ðmðvÞ þ 2qðvÞÞdðbðvÞ; b ðvÞÞ
Example 3.3. Let A ¼ ½0; ∞Þ with the usual ordering and ϕ ¼ ½0; 1; also ξ a sigma algebra of Lebesgue measurable subset of [0, 1]. We define a mapping d : ϕ A → A by
'
Since mðvÞ þ 2qðvÞ < 1 then we have bðvÞ ¼ b ðvÞ, for each v 2 ϕ. '
On the contrary, choose an arbitrary random variable b0 : ϕ → A, for '
each v 2 ϕ, there is cðvÞ 2 A which is comparable to b0 ðvÞ and b 0 ðvÞ simultaneously. If we define
dðaðvÞ; bðvÞÞ ¼ jaðvÞ bðvÞj:
c0 ðvÞ ¼ cðvÞ; cn ðvÞ ¼ Tðv; cn1 ðvÞÞ; v 2 ϕ; n 2 ℕ :
random operator T : ϕ A → A by Tðc;aÞ ¼ 1v4 þx. We define a sequence
1þ 1 2 =n for every v 2 ϕ and of mappings b : ϕ → A as b ðvÞ ¼ ð1 v2 Þ
Then ðA; d; Þ is a complete partially ordered metric spaces. Define 2
Then bn ðvÞ is comparable to cn ðvÞ, for each v 2 ϕ. Hence,
n
dðbn ðvÞcn ðvÞÞ dðTðv; bn1 ðvÞÞ; Tðv; cn1 ðvÞÞÞ mðvÞdðbn1 ðvÞ; cn1 ðvÞÞ þnðvÞ½dðbn1 ðvÞ; Tðv; bn1 ðvÞÞÞ þ dðcn1 ðvÞ; Tðv; cn1 ðvÞÞÞ þqðvÞ½dðbn1 ðvÞ; Tðv; cn1 ðvÞÞÞ þ dðcn1 ðvÞ; Tðv; bn1 ðvÞÞÞ
Remark 3.4. Theorem 3.1 is an extension of the result of Saluja and Tripathi [8] (Corollary 1) in the setting of partially ordered metric spaces. If bðvÞ ¼ cðvÞ ¼ 0 in Theorem 3.1, then we obtain the result of Nieto et al. [9] Theorem 4.1. Corollary 3.5. Let ðϕ; TÞ be a measurable space and ðA; d; αÞ be a complete partially ordered separable metric space. If T : ϕ A → A is a continuous random operator such that, Tðv; Þ is a monotone (either order-preserving or order-reserving) operator. Suppose that the following conditions hold: (J1) For each v 2 ϕ, there exists 0 mðvÞ < 1 such that
¼ mðvÞdðbn1 ðvÞ; cn1 ðvÞÞ þnðvÞ½dðbn1 ðvÞ; bn ðvÞÞ þ dðcn1 ðvÞ; cn ðvÞÞ þqðvÞ½dðbn1 ðvÞ; cn ðvÞÞ þ dðcn1 ðvÞ; bn ðvÞÞ mðvÞdðbn1 ðvÞ; cn1 ðvÞÞ þnðvÞ½dðbn1 ðvÞ; bn ðvÞÞ þ dðcn1 ðvÞ; bn ðvÞÞ þ dðbn ðvÞ; cn ðvÞÞ þqðvÞ½dðbn1 ðvÞ; cn ðvÞÞ þ dðcn1 ðvÞ; bn ðvÞÞ:
dðTðv; a1 Þ; Tðv; a2 ÞÞ mðvÞdða1 ; a2 Þ: For each a1 ; a2 2 A; a1 a2 : (J2) There is a random variable a0 : ϕ → A with
As n → ∞, we have
a0 ðvÞ Tðv; a0 ðvÞÞ; for all v 2 ϕ;
dðbðvÞ; cn ðvÞÞ mðvÞdðbðvÞ; cn1 ðvÞÞ þnðvÞ½dðcn1 ðvÞ; bðvÞÞ þ dðbðvÞ; cn ðvÞÞ þqðvÞ½dðbðvÞ; cn ðvÞÞ þ dðcn1 ðvÞ; bðvÞÞ
Or a0 ðvÞ Tðv; a0 ðvÞÞ; for all v 2 ϕ:
ðmðvÞ þ nðvÞ þ qðvÞÞdðbðvÞ; cn1 ðvÞÞ þðnðvÞ þ qðvÞÞdðbðvÞ; cn ðvÞÞ
n
n 2 ℕ. Define a measurable mapping b : ϕ → A as bðvÞ ¼ ð1 v2 Þ for each v 2 ϕ. Thus ð1 v2 Þ is the random fixed point of the random operator.
Then there is a random variable a : ϕ → A which is a random fixed point of T. Corollary 3.6. Let ðϕ; TÞ be a measurable space and ðA; d; αÞ be a complete partially ordered separable metric space. If T : ϕ A → A is a continuous random operator such that, Tðv; Þ is a monotone (either
mðvÞ þ nðvÞ þ qðvÞ dðbðvÞ; cn1 ðvÞÞ: 1 nðvÞ qðvÞ
3
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continuous linear operator from CðS; L2 ðϕ; ξ; μÞÞ into itself.
order-preserving or order-reserving) operator. Suppose that the following conditions hold: (J1) For each v 2 ϕ, there exists mðvÞ 2 0; 12 such that
Definition 4.2. [7]: Let E and F be Banach spaces. The pair (E, F) is said to be admissible with respect to a random operator, ½ XðvÞaðt; vÞ if ½ XðvÞaðt; vÞðEÞ⊂F.
dðTðv; a1 Þ; Tðv; a2 ÞÞ mðvÞ½dða1 ; Tðv; a1 ÞÞ þ dða2 ; Tðv; a2 ÞÞ:
Lemma 4.3. [7]: If (A) is a continuous linear operator from CðS; L2 ðϕ; ξ; μÞÞ into itself and E; F⊂CðS; L2 ðϕ; ξ; μÞÞ are Banach spaces stronger than CðS; L2 ðϕ; ξ; μÞÞ such that (E, F) is admissible with respect to A, then (A) is continuous from E to F. The following theorem gives the existence and uniqueness of a random solution of Eq. (4).
For each a1 ; a2 2 A; a1 a2 : (J2) There is a random variable a0 : ϕ → A with a0 ðvÞ Tðv; a0 ðvÞÞ; for all v 2 ϕ; Or
Theorem 4.1. Let the random integral Eq. (4) be subjected to the following conditions:
a0 ðvÞ Tðv; a0 ðvÞÞ; for all v 2 ϕ: Then there is a random variable a : ϕ → A which is a random fixed point of T.
(i) E and F are Banach spaces stronger than CðS; L2 ðϕ; ξ; μÞÞ such that (E, F) is admissible with respect to the integral operator defined by (4). (ii) aðt; vÞ → Tðt; aðt; vÞÞ is an operator from the set
4. Discussion
UðαÞ ¼ aðt; vÞ : aðt; vÞ 2 F; aðt; vÞ jF α
The theory of random nonlinear integral equations play a significant role in modeling physical phenomena in various branches of mathematics and applied sciences. Many mathematicians have established the existence and uniqueness of a solution for random integral equations via fixed point theorems. For instance, see [3]and [11]. To prove the existence and uniqueness of the solution of random nonlinear integral equation presented as follows Z aðt; vÞ ¼ hðt; vÞ þ
kðt; s; vÞTðs; aðs; vÞÞdμðsÞ
into the space E satisfying Tðt; aðt; vÞÞ Tðt; bðt; vÞÞ j mðvÞ aðt; vÞ bðt; vÞ j E E þnðvÞ½ aðt; vÞ Tðt; aðt; vÞÞ jE þ bðt; vÞ Tðt; bðt; vÞÞ jE þqðvÞ½ aðt; vÞ Tðt; bðt; vÞÞ jE þ bðt; vÞ Tðt; aðt; vÞÞ jE a:s: For aðt; vÞ; bðt; vÞ 2 UðαÞ, with mðvÞ þ qðvÞ < 1 , nðvÞ < 1 and
(4)
S
(iii) hðt; vÞ 2 F.
We apply Theorem 3.1 and Theorem 3.2 to obtain the result. The following assumptions are made with respect to the random kernel kðt;s; vÞ. Let S be a locally compact metric space with metric defined on S S and let μ be a complete σ - finite measure defined on Borel subset of S. Suppose S is a countable family of compact subset fcn g having the properties that c1 ⊂c2 ⊂⋯ and for any other compact set S there is a ci contain in it [10].
Then (4) has a unique random solution in UðαÞ provided kðvÞ < 1 a.s. and hðt; vÞ j þ 2kðvÞ Tðt; 0Þ j αð1 kðvÞÞa:s:where kðvÞ is the norm F E of XðvÞ. Proof: For arbitrary a0 ðt; vÞ 2 UðαÞ, we choose a1 ðt; vÞ 2 UðαÞ such that a0 ðt; vÞ ¼ Tðt; a0 ðt; vÞÞ ¼ a1 ðt; vÞ. This shows that a0 ðt; vÞis the solution of the operator Y(v). Assume a0 ðt; vÞ Tðt; a0 ðt; vÞÞwe have a0 ðt; vÞ Tðt; a0 ðt; vÞÞ ¼ a1 ðt; vÞ. For a2 ðt; vÞ 2 UðαÞ we have a1 ðt; vÞ T 2 ðt; a0 ðt; vÞÞ ¼ a2 ðt; vÞ. Continue the process we have a0 ðt; vÞ Tðt; a0 ðt; vÞÞ ¼ a1 ðt; vÞ T 2 ðt; a0 ðt; vÞÞ ¼ a2 ðt; vÞ ⋯ T n ðt; a0 ðt; vÞÞ ¼ ⋯Thus we define a sequence T n ðt; a0 ðt; vÞÞ 2 UðαÞ such that T n ðt; a0 ðt; vÞÞ T nþ1 ðt; a0 ðt; vÞÞ. This shows that it is monotone nondecreasing. Taking a0 ðt; vÞ; b0 ðt; vÞ 2 UðαÞ; a0 ðt; vÞ < b0 ðt; vÞwe define a metric d on UðαÞ by ja0 ðt; vÞ b0 ðt; vÞj. This shows that ðUðαÞ; d; Þ is a complete partially ordered separable metric space. Next we prove that the random nonlinear integral operator is contractive. Suppose the operator YðvÞ from UðαÞ into F is defined by
Definition 4.1. [7]: Let CðS; L2 ðϕ; ξ; μÞÞ be the space of all continuous function from S into L2 ðϕ; ξ; μÞ with the topology of uniform convergence on compact set S, that is, for each fixed t 2 S; aðt; vÞ is random variable R such that jjaðt; vÞjj2L2ðϕ;ξ;μÞ ¼ aðt; vÞj2 dμðvÞ < ∞: ϕ
Consider the function hðt; vÞand Tðt; aðt; vÞÞ to be in the space CðS; L2 ðϕ; ξ; μÞÞ concerning the random kernel. We assume that for ðt; sÞ; kðt; s; vÞ 2 L∞ ðϕ; ξ; μÞ the norm is denoted by kðt; s; vÞ ¼ kðt; s; vÞ jL∞ðϕ;ξ;μÞ ¼ μ ess sup kðt; s; vÞ : v2ϕ Suppose we assume kðt; s; vÞ to be kðt; s; vÞ : aðt; vÞ jL2ðϕ;ξ;μÞ isμ-
Z
integrable with respect to S for each t 2 S and aðs; vÞ in CðS; L2 ðϕ; ξ; μÞÞ and there is a function J which defined μ a. e. on S such that JðsÞ aðs; vÞ jL2ðϕ;ξ;μÞ is μ- integrable and for ðt; sÞ 2 S S, JðuÞ aðt; vÞ jL2ðϕ;ξ;μÞ μ kðt; u; vÞ kðs; u; vÞ : aðt; vÞ jL2ðϕ;ξ;μÞ
½YðvÞaðt; vÞ ¼ hðt; vÞ þ
Then we obtain the following from the conditions of the theorem. ½YðvÞaðt; vÞ j F
a. e. Therefore for ðt;sÞ 2 S S, we obtain kðt;s; vÞ ðs; vÞ 2 L2 ðϕ;ξ; μÞ. We now define the random integral operator Z ½XðvÞaðt; vÞ ¼
kðt; s; vÞTðs; aðs; vÞÞdμðsÞ:
S
hðt; vÞ jF þ kðvÞ Tðt; aðt; vÞÞ jD a:s: þ kðvÞ Tðt; 0Þ hðt; vÞ j E jE þ kðvÞ Tðt; aðt; vÞ Tðt; 0Þ jE :
But, aðt; vÞ 0 j kðvÞ Tðt; E aðt; vÞ Tðt; 0Þ jE kðvÞðmðvÞ þnðvÞ½ aðt; vÞ Tðt; aðt; vÞÞ jE þ 0 Tðt; 0Þ jE þ qðvÞ½aðt; vÞ Tðt; 0Þ j þ 0 Tðt; aðt; vÞÞ j Þ
kðt; s; vÞaðs; vÞdμðsÞ:
S
Where the integral is a Bochner integral. From the conditions on kðt;s; vÞ, we obtain that for each t 2 S; ½ XðvÞaðt; vÞ 2 L2 ðϕ; ξ; μÞ and XðvÞ is a
E
4
E
K.S. Eke et al.
Heliyon 5 (2019) e01641
aðt; vÞ j kðvÞðmðvÞ E þnðvÞ½ aðt; vÞ Tðt; aðt; vÞÞ jE þ Tðt; 0Þ þ qðvÞ½ aðt; vÞ j þ Tðt; aðt; vÞÞ j Þ
solution for nonlinear integral equation is established using this contractive operator.
kðvÞ½ðmðvÞ þ qðvÞÞ aðt; vÞ jE þ nðvÞ Tðt; 0Þ jE
Author contribution statement
E
E
Declarations
K.S. Eke; Wrote the paper. H. Akewe; Conceived and designed the experiment. S.A. Bishop; Analyzed and interpreted the data.
kðvÞα þ kðvÞ Tðt; 0Þ jE ; Since mðvÞ þ qðvÞ < 1 and nðvÞ < 1. Hence, ½YðvÞaðt; vÞ j hðt; vÞ j þ kðvÞ Tðt; 0Þ j F E F þkðvÞ Tðt; 0Þ α þ kðvÞ jE hðt; vÞ jE þ 2kðvÞ Tðt; 0Þ jE þ kðvÞαa:s: αð1 kðvÞÞ þ kðvÞα a:s: < α:
Funding statement This work was supported by Covenant University. Competing interest statement
Therefore, ½YðvÞaðt; vÞ 2 UðαÞ. If aðt; vÞ; bðt; vÞ 2 UðαÞ then using condition (ii) we obtain,
The authors declare no conflict of interest.
½YðvÞaðt; vÞ ½YðvÞbðt; vÞ j F 3 Z ¼ kðt; s; vÞ½Tðs; aðs; vÞ Tðs; bðs; vÞ5dμðsÞ jE S kðvÞ Tðt; aðt; vÞ Tðt; bðt; vÞÞ jE a:s::
Additional information No additional information is available for this paper. References
aðt; vÞ bðt; vÞ j kðvÞðmðvÞ E þnðvÞ½ aðt; vÞ Tðt; aðt; vÞÞ jE þ bðt; vÞ Tðt; bðt; vÞÞ jE þqðvÞ½aðt; vÞ Tðt; bðt; vÞÞ jE þ bðt; vÞ Tðt; aðt; vÞÞ jE Þ a:s:
[1] S. Banach, Sur les operations dans les ensembles abstraits et leur applications aux equations integrales, Fundam. Math. 3 (1922) 133–181. [2] A.C.M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Am. Math. Soc. 132 (2004) 1435–1443. [3] R.A. Rashwan, D.M. Albaqeri, A common fixed point theorem and application to random integral equations, Int. J. Appl. Math. Res. 3 (1) (2014) 71–80. [4] O. Hans, Random fixed point theorems, in: 1957 Transactions of the First Prague Conference on Information Theory, Statistical Decision Functions, Random Processses Held at Liblice Near Prague from November 28 to 30, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1956, pp. 105–125. [5] O. Hans, A. Spacek, Randon fixed point approximation by differentiable trajectories. Trans. 2nd Prague Conf. Information Theory, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1960, pp. 203–213. [6] G.A. Okeke, K.S. Eke, Convergence and almost sure T-stability for random Noortype iterative scheme, Int. J. Pure Appl. Math. 107 (1) (2016) 1–16. [7] C.H. Lee, W.J. Padgett, On random nonlinear contractions, Math. Syst. Theor. 11 (1977) 77–84. [8] G.S. Saluja, B.P. Tripathi, Some common random fixed point theorems for contractive type conditions in cone random metric spaces, Acta Univ. Sapientiae, Math. 8 (1) (2016) 174–190. [9] J.J. Nieto, A. Ouahab, R. Rodriguez-Lopez, Random fixed point theorems in partially ordered metric spaces, Fixed Point Theory Appl. 98 (2016) 2016. [10] M.C. Joshi, R.K. Bose, Some Topics in Nonlinear Functional Analysis, Wiley, New York, 1984. [11] W. Sintunavarat, Fixed point results in b-metric spaces approach to the existence of a solution for nonlinear integral equations, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 110 (2) (2016) 585–600.
mðvÞ aðt; vÞ bðt; vÞ jE aðt; vÞ Tðt; aðt; vÞÞ j þnðvÞ½ E þ bðt; vÞ Tðt; bðt; vÞÞ jE þqðvÞ½aðt; vÞ Tðt; bðt; vÞÞ jE þ bðt; vÞ Tðt; aðt; vÞÞ j a:s: E
Since kðvÞ < 1 a.s. Therefore Y(v) is a random nonlinear contraction operator on UðαÞ. By Theorem 3.1 and 3.2, there is a random variable y : ϕ → B which is the unique random solution of the operator (4). 5. Conclusion This research proved the existence and uniqueness of random fixed point for certain contractive mappings in complete partially ordered separable metric spaces. The existence and uniqueness of a random
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On random fixed point theorems with applications to integral equations Kanayo Stella Eke *, Hudson Akewe, Sheila Amina Bishop Department of Mathematics, Covenant University, P. M. B. 1023, Ota, Ogun State, Nigeria
A R T I C L E I N F O
A B S T R A C T
Keyword: Applied mathematics
This particular research establishes some random fixed point theorems for general nonlinear random contractive operators in the context of partially ordered separable metric spaces. The existence and uniqueness of the random solution for the nonlinear integral equation is obtained by applying the result of the random fixed point. The results generalize and improve on some related works in the literature.
Our theorems are proved in the context of metric space while Saluja and Tripathi [8] proved in the context of partial metric spaces. Nieto, Ouahab and. Rodriguez-Lopez [9] proved their theorem using Banach contraction mappings while we proved our theorems using Hardy and Rogers contraction mappings. Rashwan and Albaqeri [3] proved the solution of the random integral equation using the Banach contraction operator while we proved our solution to the random integral equations employing more general contractive operator.
1. Introduction FIXED point theory plays very fundamental role in solving deterministic operator equations. An important principle that is required to solve applied mathematical problems in applied mathematics, such as computer sciences involve looping. Fixed point iteration and monotonous iteration techniques introduced by Banach [1] and Ran and Reurings [2] respectively handled these problems. Probabilistic fundamental analysis is an important aspect of mathematics that is applied to solving problems. An equation that requires a mathematical model to describe its phenomena is classified as random equation. Fixed point theorems for stochastic functions was pioneered by Prague School of Probability. In 1955–1965, Spacek and Hans studied the Fredholm integral equations with respect to random kernel. Stochastic fixed point theorem for contractive mappings in Polish spaces were established by Rashwan and Albaqeri [3], Hans [4], Hans and Spacek [5], Okeke and Eke [6]. In 1977, Lee and Padgett [7] proved fixed point result for stochastic nonlinear contractive mappings in separable Banach spaces and applied the result obtained to establish the existence and uniqueness of the random solution of the stochastic nonlinear integral equations. Saluja and Tripathi [8] proved some stochastic fixed point results for contractive mappings in the framework of cone random metric spaces.
Nieto et al. [9] established a stochastic fixed point theorem for Banach contraction mappings in ordered metric spaces. In the same reference, the result obtained is used to established the solution for random differential equations with boundary properties. This particular paper proves the existence and uniqueness of random fixed point for a more general nonlinear contractive functions in partially ordered separable metric spaces. The result obtain is apply to establish the existence and uniqueness of random solution for nonlinear integral equation. 2. Methodology Suppose ðA; ξÞ is a separable Banach space, where ξ is aσ -algebra of Borel subsets of A and ðϕ; ξ; μÞ represents a complete probability measure space with measureμ. ξ is also a σ -algebra of subsets of ϕ. The following definitions are found in Joshi and Bose [10] with more details for the readers. Definition 1.1. An operator T : ϕ A → B is known as a random operator if Tðv; aÞ ¼ BðvÞ is a random variable for every a 2 A. Definition 1.2. An operator T : ϕ A → B is called continuous random mapping if the set of all v 2 ϕ for which Tðv; aÞ is a continuous function of
* Corresponding author. E-mail address: [email protected] (K.S. Eke). https://doi.org/10.1016/j.heliyon.2019.e01641 Received 17 October 2018; Received in revised form 29 April 2019; Accepted 30 April 2019 2405-8440/© 2019 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article as: K.S. Eke et al., On random fixed point theorems with applications to integral equations, Heliyon, https://doi.org/10.1016/ j.heliyon.2019.e01641
K.S. Eke et al.
Heliyon 5 (2019) e01641
a has measure one.
Definition 1.3. An equation of the type Tðv; aðvÞÞ ¼ aðvÞ, where T : ϕ A → A is a random mapping is called a random fixed point equation.
mðvÞ þ nðvÞ þ qðvÞ dðbn1 ðvÞ; bn ðvÞÞ 1 nðvÞ qðvÞ
kdðbn1 ðvÞ; bn ðvÞÞ:
Definition 1.4. Any mapping a : ϕ → A which satisfies random fixed point equation Tðv; aðvÞÞ ¼ aðvÞ almost surely is called a wide sense solution of the fixed point equation.
Where k ¼
mðvÞþnðvÞþqðvÞ 1nðvÞqðvÞ
< 1:
Consequently, we get
Definition 1.5. Any A-valued random variable aðvÞ which satisfies μfv : Tðv; aðvÞÞ ¼ aðvÞg ¼ 1 is called random solution of the fixed point equation. Let A be a nonempty set and let T : A → A be an operator. We define the Picard iteration of T by T n ¼ TðT n1 Þ; for n 2 ℕ; n 2:
dðbn ðvÞ; bnþ1 ðvÞÞ ½kðvÞn dðb0 ðvÞ; b1 ðvÞÞ: For n > m we obtain, dðbm ðvÞ; bn ðvÞÞ dðbn ðvÞ; bmþ1 ðvÞÞ þ dðbmþ1 ðvÞ; bmþ2 ðvÞÞ þ ⋯ þ dðbn1 ðvÞ; bn ðvÞÞ
Definition 1.6. [9]: If ðA; Þis a partially ordered set and T : A → A , we say that T is monotone non decreasing if a b;a;b 2 A ⇒ TðaÞ TðbÞ.
ð½kðvÞm þ ½kðvÞ
3. Results
mþ1
This section proves the existence and uniqueness of random fixed point for contractive mappings in partially ordered separable metric spaces.
mþn1 þ ⋯ þ kðvÞ Þdðb0 ðvÞ; b1 ðvÞ
½kðvÞm dðb0 ðvÞ; b1 ðvÞÞ 1 kðvÞ
This shows that fbn ðvÞgn2ℕ is a Cauchy sequence for every v 2 ϕ. Let bðvÞ be the limit point of fbn ðvÞg, v 2 ϕ. Since b0 ðvÞ is measurable, then b1 ðvÞ is measurable. Consequently, for each n 2 ℕ, the operator v → bn ðvÞ is measurable. This implies that bðvÞ is measurable. Next we prove that bðvÞ is the random fixed point of T.
Theorem 3.1. Let ðϕ; TÞ be a measurable space and ðA; d; αÞ a complete partially ordered separable metric space. If T : ϕ A → A is a continuous random operator such that, Tðv; Þ is a monotonous operator. If the following conditions hold: (J1) For each v 2 ϕ, there exists mðvÞ þnðvÞ þ 2qðvÞ < 1 such that dðTðv; a1 Þ; Tðv; a2 ÞÞ mðvÞdða1 ; a2 Þ þnðvÞ½dða1 ; Tðv; a1 ÞÞ þ dða2 ; Tðv; a2 ÞÞ þqðvÞ½dða1 ; Tðv; a2 ÞÞ þ dða2 ; Tðv; a1 ÞÞ:
(2)
dðbðvÞ; Tðv; bðvÞÞ dðbðvÞ; bn ðvÞÞ þ dðbn ðvÞ; Tðv; bðvÞÞ dðbðvÞ; bn ðvÞÞ þ dðTðv; bn1 ðvÞÞ; Tðv; bðvÞÞÞ dðbðvÞ; bn ðvÞÞ þ mðvÞdðbn1 ðvÞ; bðvÞÞ þ nðvÞ½dðbn1 ðvÞ; Tðv; bn1 ðvÞÞ þ dðbðvÞ; Tðv; bðvÞÞ þ qðvÞ½dðbn1 ðvÞ; Tðv; bðvÞÞ þ dðbðvÞ; Tðv; bn1 ðvÞÞ:
(1)
As n → ∞, we get
For each a1 ; a2 2 A; a1 a2 : (J2) There is a random variable a0 : ϕ → A with
dðbðvÞ; Tðv; bðvÞÞ dðbðvÞ; ðvÞÞ þ mðvÞdðbðvÞ; bðvÞÞ þ nðvÞ½dðbðvÞ; Tðv; bðvÞÞ þ dðbðvÞ; Tðv; bðvÞÞ þ qðvÞ½dðbðvÞ; Tðv; bðvÞÞ þ dðbðvÞ; Tðv; bðvÞÞ
a0 ðvÞ Tðv; a0 ðvÞÞ; for all v 2 ϕ; or
ð2nðvÞ þ 2qðvÞÞdðbðvÞ; Tðv; bðvÞÞ
a0 ðvÞ Tðv; a0 ðvÞÞ; for all v 2 ϕ:
Since 2nðvÞ þ 2qðvÞ < 1 then it is a contradiction. Therefore bðvÞ is the random fixed point of T. To prove the uniqueness of the random fixed point of T, we consider the following proposition in [2]. (J3) Every pair of elements of A has a lower bound or upper bound. In [2], the above condition [J3] is equivalent to: [J3’] for every a; b 2 A, there is c 2 A that is comparable to a and b. The following theorem concludes that the existence of a random variable b : ϕ → A in Theorem 3.1 is the unique random fixed point of T.
Then there is a random variable a : ϕ → A which is a random fixed point of T. Proof: If for v 2 ϕ, there is a random variable a0 ðvÞ 2 ðϕ; TÞsuch that Tðv; a0 ðvÞÞ ¼ a0 ðvÞthen a0 ðvÞis a random fixed point of T. On the contrary, we assume that Tðv; a0 ðvÞÞ 6¼ a0 ðvÞ for some v 2 ϕ. Let b0 ðvÞ ¼ a0 ðvÞ, then we can define a sequence Tðv;bn1 ðvÞÞ ¼ bn ðvÞ, for v 2 ϕ, n 2 ℕ. According to (J1) we have, for each v 2 ϕ, and n 2 ℕ one of the following relations hold: bn ðvÞ bnþ1 ðvÞ or bn ðvÞ bnþ1 ðvÞ. Using the above inequality in Eq. (1) we obtain,
Theorem 3.2. Let all the hypotheses of Theorem 3.1 be satisfied and if the following condition [J3] (equivalently [J3’]) be satisfied, then b : ϕ → A is the unique random fixed point of T.
dðbn ðvÞ; bnþ1 ðvÞÞ dðTðv; bn1 ðvÞÞ; Tðv; bn ðvÞÞÞ mðvÞdðbn1 ðvÞ; bn ðvÞÞ þnðvÞ½dðbn1 ðvÞ; Tðv; bn1 ðvÞÞÞ þ dðbn ðvÞ; Tðv; bn ðvÞÞÞ þqðvÞ½dðbn1 ðvÞ; Tðv; bn ðvÞÞÞ þ dðbn ðvÞ; Tðv; bn1 ðvÞÞÞ
'
Proof: Let b : ϕ → A be an arbitrary random variable and we define the sequence ' ' ' ' b 0 ðvÞ ¼ a 0 ðvÞ; bn ðvÞ ¼ T v; b n1 ðvÞ ; v 2 ϕ; n 2 ℕ:
mðvÞdðbn1 ðvÞ; bn ðvÞÞ þnðvÞ½dðbn1 ðvÞ; bn ðvÞÞ þ dðbn ðvÞ; bnþ1 ðvÞÞ þqðvÞ½dðbn1 ðvÞ; bnþ1 ðvÞÞ þ dðbn ðvÞ; bn ðvÞÞ
In Theorem 3.1, we obtain that fbn ðvÞgn2ℕ → bðvÞ as n → ∞, for each '
v 2 ϕ, where bðvÞ is the random fixed point of T. If b 0 ðvÞ is comparable to
mðvÞdðbn1 ðvÞ; bn ðvÞÞ þnðvÞ½dðbn1 ðvÞ; bn ðvÞÞ þ dðbn ðvÞ; bnþ1 ðvÞÞ þqðvÞ½dðbn1 ðvÞ; bn ðvÞÞ þ dðbn ðvÞ; bnþ1 ðvÞÞ
b0 ðvÞ for each v 2 ϕ, it follows that Tðv; b0 ðvÞÞ' is comparable to Tðv; b0 ðvÞÞ '
for each v 2 ϕ. Therefore b n ðvÞ is comparable to bn ðvÞ for each v 2 ϕ. Hence,
ðmðvÞ þ nðvÞ þ qðvÞÞdðbn1 ðvÞ; bn ðvÞÞ þ ðnðvÞ þ qðvÞÞdðbn ðvÞ; bnþ1 ðvÞÞ
2
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' ' d bn ðvÞ;b n ðvÞ d Tðv;bn1 ðvÞÞ;T v;b n1 ðvÞ ' mðvÞd bn1 ðvÞ;b n1 ðvÞ
Similarly, we obtain
mðvÞ þ nðvÞ þ qðvÞ
' ' d bðvÞ; c n ðvÞ d bðvÞ; c n1 ðvÞ : 1 nðvÞ qðvÞ
þnðvÞ½dðb n1 ðvÞ;Tðv;b n1 ðvÞÞÞ ' ' þd b n1 ðvÞ;T v;b n1 ðvÞ ' þqðvÞ d bn1 ðvÞ;T v;b n1 ðvÞ ' þd b n1 ðvÞ;Tðv;bn1 ðvÞÞ
Choose a natural number N1 such that dðbðvÞ; cn ðvÞÞ
Therefore, ' ' d bðvÞ; b ðvÞ dðbðvÞ; cn1 ðvÞÞ þ d cn1 ðvÞ; b ðvÞ
As n → ∞, we get
mðvÞ þ nðvÞ þ qðvÞ εð1 nðvÞ qðvÞÞ 1 nðvÞ qðvÞ 2ðmðvÞ þ nðvÞ þ qðvÞÞ
mðvÞ þ nðvÞ þ qðvÞ εð1 nðvÞ qðvÞÞ þ 1 nðvÞ qðvÞ 2ðmðvÞ þ nðvÞ þ qðvÞÞ
d bðvÞ; bðvÞÞ mðvÞdðbðvÞ; b ðvÞ ' ' þ nðvÞ dðbðvÞ; bðvÞÞ þ dðb ðvÞ; b ðvÞÞ ' ' þ qðvÞ dðbðvÞ; b ðvÞÞ þ dðb ðvÞ; bðvÞÞ '
(3)
and
εð1 nðvÞ qðvÞÞ ' : d bðvÞ; c n ðvÞ 2ðmðvÞ þ nðvÞ þ qðvÞÞ
' ¼ mðvÞd bn1 ðvÞ; b n1 ðvÞ ' ' þnðvÞ dðbn1 ðvÞ; bn ðvÞÞ þ d b n1 ðvÞ; b n ðvÞ ' ' þqðvÞ d bn1 ðvÞ; b n ðvÞ þ d b n1 ðvÞ; bn ðvÞ :
εð1 nðvÞ qðvÞÞ 2ðmðvÞ þ nðvÞ þ qðvÞÞ
'
¼
ε 2
ε
þ ¼ ε: 2
Since ε is the smallest positive number we assumeε ¼ 0. Therefore '
'
'
dðbðvÞ; b ðvÞÞ ¼ 0. Thus bðvÞ ¼ b ðvÞ: The uniqueness proved.
ðmðvÞ þ 2qðvÞÞdðbðvÞ; b ðvÞÞ
Example 3.3. Let A ¼ ½0; ∞Þ with the usual ordering and ϕ ¼ ½0; 1; also ξ a sigma algebra of Lebesgue measurable subset of [0, 1]. We define a mapping d : ϕ A → A by
'
Since mðvÞ þ 2qðvÞ < 1 then we have bðvÞ ¼ b ðvÞ, for each v 2 ϕ. '
On the contrary, choose an arbitrary random variable b0 : ϕ → A, for '
each v 2 ϕ, there is cðvÞ 2 A which is comparable to b0 ðvÞ and b 0 ðvÞ simultaneously. If we define
dðaðvÞ; bðvÞÞ ¼ jaðvÞ bðvÞj:
c0 ðvÞ ¼ cðvÞ; cn ðvÞ ¼ Tðv; cn1 ðvÞÞ; v 2 ϕ; n 2 ℕ :
random operator T : ϕ A → A by Tðc;aÞ ¼ 1v4 þx. We define a sequence
1þ 1 2 =n for every v 2 ϕ and of mappings b : ϕ → A as b ðvÞ ¼ ð1 v2 Þ
Then ðA; d; Þ is a complete partially ordered metric spaces. Define 2
Then bn ðvÞ is comparable to cn ðvÞ, for each v 2 ϕ. Hence,
n
dðbn ðvÞcn ðvÞÞ dðTðv; bn1 ðvÞÞ; Tðv; cn1 ðvÞÞÞ mðvÞdðbn1 ðvÞ; cn1 ðvÞÞ þnðvÞ½dðbn1 ðvÞ; Tðv; bn1 ðvÞÞÞ þ dðcn1 ðvÞ; Tðv; cn1 ðvÞÞÞ þqðvÞ½dðbn1 ðvÞ; Tðv; cn1 ðvÞÞÞ þ dðcn1 ðvÞ; Tðv; bn1 ðvÞÞÞ
Remark 3.4. Theorem 3.1 is an extension of the result of Saluja and Tripathi [8] (Corollary 1) in the setting of partially ordered metric spaces. If bðvÞ ¼ cðvÞ ¼ 0 in Theorem 3.1, then we obtain the result of Nieto et al. [9] Theorem 4.1. Corollary 3.5. Let ðϕ; TÞ be a measurable space and ðA; d; αÞ be a complete partially ordered separable metric space. If T : ϕ A → A is a continuous random operator such that, Tðv; Þ is a monotone (either order-preserving or order-reserving) operator. Suppose that the following conditions hold: (J1) For each v 2 ϕ, there exists 0 mðvÞ < 1 such that
¼ mðvÞdðbn1 ðvÞ; cn1 ðvÞÞ þnðvÞ½dðbn1 ðvÞ; bn ðvÞÞ þ dðcn1 ðvÞ; cn ðvÞÞ þqðvÞ½dðbn1 ðvÞ; cn ðvÞÞ þ dðcn1 ðvÞ; bn ðvÞÞ mðvÞdðbn1 ðvÞ; cn1 ðvÞÞ þnðvÞ½dðbn1 ðvÞ; bn ðvÞÞ þ dðcn1 ðvÞ; bn ðvÞÞ þ dðbn ðvÞ; cn ðvÞÞ þqðvÞ½dðbn1 ðvÞ; cn ðvÞÞ þ dðcn1 ðvÞ; bn ðvÞÞ:
dðTðv; a1 Þ; Tðv; a2 ÞÞ mðvÞdða1 ; a2 Þ: For each a1 ; a2 2 A; a1 a2 : (J2) There is a random variable a0 : ϕ → A with
As n → ∞, we have
a0 ðvÞ Tðv; a0 ðvÞÞ; for all v 2 ϕ;
dðbðvÞ; cn ðvÞÞ mðvÞdðbðvÞ; cn1 ðvÞÞ þnðvÞ½dðcn1 ðvÞ; bðvÞÞ þ dðbðvÞ; cn ðvÞÞ þqðvÞ½dðbðvÞ; cn ðvÞÞ þ dðcn1 ðvÞ; bðvÞÞ
Or a0 ðvÞ Tðv; a0 ðvÞÞ; for all v 2 ϕ:
ðmðvÞ þ nðvÞ þ qðvÞÞdðbðvÞ; cn1 ðvÞÞ þðnðvÞ þ qðvÞÞdðbðvÞ; cn ðvÞÞ
n
n 2 ℕ. Define a measurable mapping b : ϕ → A as bðvÞ ¼ ð1 v2 Þ for each v 2 ϕ. Thus ð1 v2 Þ is the random fixed point of the random operator.
Then there is a random variable a : ϕ → A which is a random fixed point of T. Corollary 3.6. Let ðϕ; TÞ be a measurable space and ðA; d; αÞ be a complete partially ordered separable metric space. If T : ϕ A → A is a continuous random operator such that, Tðv; Þ is a monotone (either
mðvÞ þ nðvÞ þ qðvÞ dðbðvÞ; cn1 ðvÞÞ: 1 nðvÞ qðvÞ
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continuous linear operator from CðS; L2 ðϕ; ξ; μÞÞ into itself.
order-preserving or order-reserving) operator. Suppose that the following conditions hold: (J1) For each v 2 ϕ, there exists mðvÞ 2 0; 12 such that
Definition 4.2. [7]: Let E and F be Banach spaces. The pair (E, F) is said to be admissible with respect to a random operator, ½ XðvÞaðt; vÞ if ½ XðvÞaðt; vÞðEÞ⊂F.
dðTðv; a1 Þ; Tðv; a2 ÞÞ mðvÞ½dða1 ; Tðv; a1 ÞÞ þ dða2 ; Tðv; a2 ÞÞ:
Lemma 4.3. [7]: If (A) is a continuous linear operator from CðS; L2 ðϕ; ξ; μÞÞ into itself and E; F⊂CðS; L2 ðϕ; ξ; μÞÞ are Banach spaces stronger than CðS; L2 ðϕ; ξ; μÞÞ such that (E, F) is admissible with respect to A, then (A) is continuous from E to F. The following theorem gives the existence and uniqueness of a random solution of Eq. (4).
For each a1 ; a2 2 A; a1 a2 : (J2) There is a random variable a0 : ϕ → A with a0 ðvÞ Tðv; a0 ðvÞÞ; for all v 2 ϕ; Or
Theorem 4.1. Let the random integral Eq. (4) be subjected to the following conditions:
a0 ðvÞ Tðv; a0 ðvÞÞ; for all v 2 ϕ: Then there is a random variable a : ϕ → A which is a random fixed point of T.
(i) E and F are Banach spaces stronger than CðS; L2 ðϕ; ξ; μÞÞ such that (E, F) is admissible with respect to the integral operator defined by (4). (ii) aðt; vÞ → Tðt; aðt; vÞÞ is an operator from the set
4. Discussion
UðαÞ ¼ aðt; vÞ : aðt; vÞ 2 F; aðt; vÞ jF α
The theory of random nonlinear integral equations play a significant role in modeling physical phenomena in various branches of mathematics and applied sciences. Many mathematicians have established the existence and uniqueness of a solution for random integral equations via fixed point theorems. For instance, see [3]and [11]. To prove the existence and uniqueness of the solution of random nonlinear integral equation presented as follows Z aðt; vÞ ¼ hðt; vÞ þ
kðt; s; vÞTðs; aðs; vÞÞdμðsÞ
into the space E satisfying Tðt; aðt; vÞÞ Tðt; bðt; vÞÞ j mðvÞ aðt; vÞ bðt; vÞ j E E þnðvÞ½ aðt; vÞ Tðt; aðt; vÞÞ jE þ bðt; vÞ Tðt; bðt; vÞÞ jE þqðvÞ½ aðt; vÞ Tðt; bðt; vÞÞ jE þ bðt; vÞ Tðt; aðt; vÞÞ jE a:s: For aðt; vÞ; bðt; vÞ 2 UðαÞ, with mðvÞ þ qðvÞ < 1 , nðvÞ < 1 and
(4)
S
(iii) hðt; vÞ 2 F.
We apply Theorem 3.1 and Theorem 3.2 to obtain the result. The following assumptions are made with respect to the random kernel kðt;s; vÞ. Let S be a locally compact metric space with metric defined on S S and let μ be a complete σ - finite measure defined on Borel subset of S. Suppose S is a countable family of compact subset fcn g having the properties that c1 ⊂c2 ⊂⋯ and for any other compact set S there is a ci contain in it [10].
Then (4) has a unique random solution in UðαÞ provided kðvÞ < 1 a.s. and hðt; vÞ j þ 2kðvÞ Tðt; 0Þ j αð1 kðvÞÞa:s:where kðvÞ is the norm F E of XðvÞ. Proof: For arbitrary a0 ðt; vÞ 2 UðαÞ, we choose a1 ðt; vÞ 2 UðαÞ such that a0 ðt; vÞ ¼ Tðt; a0 ðt; vÞÞ ¼ a1 ðt; vÞ. This shows that a0 ðt; vÞis the solution of the operator Y(v). Assume a0 ðt; vÞ Tðt; a0 ðt; vÞÞwe have a0 ðt; vÞ Tðt; a0 ðt; vÞÞ ¼ a1 ðt; vÞ. For a2 ðt; vÞ 2 UðαÞ we have a1 ðt; vÞ T 2 ðt; a0 ðt; vÞÞ ¼ a2 ðt; vÞ. Continue the process we have a0 ðt; vÞ Tðt; a0 ðt; vÞÞ ¼ a1 ðt; vÞ T 2 ðt; a0 ðt; vÞÞ ¼ a2 ðt; vÞ ⋯ T n ðt; a0 ðt; vÞÞ ¼ ⋯Thus we define a sequence T n ðt; a0 ðt; vÞÞ 2 UðαÞ such that T n ðt; a0 ðt; vÞÞ T nþ1 ðt; a0 ðt; vÞÞ. This shows that it is monotone nondecreasing. Taking a0 ðt; vÞ; b0 ðt; vÞ 2 UðαÞ; a0 ðt; vÞ < b0 ðt; vÞwe define a metric d on UðαÞ by ja0 ðt; vÞ b0 ðt; vÞj. This shows that ðUðαÞ; d; Þ is a complete partially ordered separable metric space. Next we prove that the random nonlinear integral operator is contractive. Suppose the operator YðvÞ from UðαÞ into F is defined by
Definition 4.1. [7]: Let CðS; L2 ðϕ; ξ; μÞÞ be the space of all continuous function from S into L2 ðϕ; ξ; μÞ with the topology of uniform convergence on compact set S, that is, for each fixed t 2 S; aðt; vÞ is random variable R such that jjaðt; vÞjj2L2ðϕ;ξ;μÞ ¼ aðt; vÞj2 dμðvÞ < ∞: ϕ
Consider the function hðt; vÞand Tðt; aðt; vÞÞ to be in the space CðS; L2 ðϕ; ξ; μÞÞ concerning the random kernel. We assume that for ðt; sÞ; kðt; s; vÞ 2 L∞ ðϕ; ξ; μÞ the norm is denoted by kðt; s; vÞ ¼ kðt; s; vÞ jL∞ðϕ;ξ;μÞ ¼ μ ess sup kðt; s; vÞ : v2ϕ Suppose we assume kðt; s; vÞ to be kðt; s; vÞ : aðt; vÞ jL2ðϕ;ξ;μÞ isμ-
Z
integrable with respect to S for each t 2 S and aðs; vÞ in CðS; L2 ðϕ; ξ; μÞÞ and there is a function J which defined μ a. e. on S such that JðsÞ aðs; vÞ jL2ðϕ;ξ;μÞ is μ- integrable and for ðt; sÞ 2 S S, JðuÞ aðt; vÞ jL2ðϕ;ξ;μÞ μ kðt; u; vÞ kðs; u; vÞ : aðt; vÞ jL2ðϕ;ξ;μÞ
½YðvÞaðt; vÞ ¼ hðt; vÞ þ
Then we obtain the following from the conditions of the theorem. ½YðvÞaðt; vÞ j F
a. e. Therefore for ðt;sÞ 2 S S, we obtain kðt;s; vÞ ðs; vÞ 2 L2 ðϕ;ξ; μÞ. We now define the random integral operator Z ½XðvÞaðt; vÞ ¼
kðt; s; vÞTðs; aðs; vÞÞdμðsÞ:
S
hðt; vÞ jF þ kðvÞ Tðt; aðt; vÞÞ jD a:s: þ kðvÞ Tðt; 0Þ hðt; vÞ j E jE þ kðvÞ Tðt; aðt; vÞ Tðt; 0Þ jE :
But, aðt; vÞ 0 j kðvÞ Tðt; E aðt; vÞ Tðt; 0Þ jE kðvÞðmðvÞ þnðvÞ½ aðt; vÞ Tðt; aðt; vÞÞ jE þ 0 Tðt; 0Þ jE þ qðvÞ½aðt; vÞ Tðt; 0Þ j þ 0 Tðt; aðt; vÞÞ j Þ
kðt; s; vÞaðs; vÞdμðsÞ:
S
Where the integral is a Bochner integral. From the conditions on kðt;s; vÞ, we obtain that for each t 2 S; ½ XðvÞaðt; vÞ 2 L2 ðϕ; ξ; μÞ and XðvÞ is a
E
4
E
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Heliyon 5 (2019) e01641
aðt; vÞ j kðvÞðmðvÞ E þnðvÞ½ aðt; vÞ Tðt; aðt; vÞÞ jE þ Tðt; 0Þ þ qðvÞ½ aðt; vÞ j þ Tðt; aðt; vÞÞ j Þ
solution for nonlinear integral equation is established using this contractive operator.
kðvÞ½ðmðvÞ þ qðvÞÞ aðt; vÞ jE þ nðvÞ Tðt; 0Þ jE
Author contribution statement
E
E
Declarations
K.S. Eke; Wrote the paper. H. Akewe; Conceived and designed the experiment. S.A. Bishop; Analyzed and interpreted the data.
kðvÞα þ kðvÞ Tðt; 0Þ jE ; Since mðvÞ þ qðvÞ < 1 and nðvÞ < 1. Hence, ½YðvÞaðt; vÞ j hðt; vÞ j þ kðvÞ Tðt; 0Þ j F E F þkðvÞ Tðt; 0Þ α þ kðvÞ jE hðt; vÞ jE þ 2kðvÞ Tðt; 0Þ jE þ kðvÞαa:s: αð1 kðvÞÞ þ kðvÞα a:s: < α:
Funding statement This work was supported by Covenant University. Competing interest statement
Therefore, ½YðvÞaðt; vÞ 2 UðαÞ. If aðt; vÞ; bðt; vÞ 2 UðαÞ then using condition (ii) we obtain,
The authors declare no conflict of interest.
½YðvÞaðt; vÞ ½YðvÞbðt; vÞ j F 3 Z ¼ kðt; s; vÞ½Tðs; aðs; vÞ Tðs; bðs; vÞ5dμðsÞ jE S kðvÞ Tðt; aðt; vÞ Tðt; bðt; vÞÞ jE a:s::
Additional information No additional information is available for this paper. References
aðt; vÞ bðt; vÞ j kðvÞðmðvÞ E þnðvÞ½ aðt; vÞ Tðt; aðt; vÞÞ jE þ bðt; vÞ Tðt; bðt; vÞÞ jE þqðvÞ½aðt; vÞ Tðt; bðt; vÞÞ jE þ bðt; vÞ Tðt; aðt; vÞÞ jE Þ a:s:
[1] S. Banach, Sur les operations dans les ensembles abstraits et leur applications aux equations integrales, Fundam. Math. 3 (1922) 133–181. [2] A.C.M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Am. Math. Soc. 132 (2004) 1435–1443. [3] R.A. Rashwan, D.M. Albaqeri, A common fixed point theorem and application to random integral equations, Int. J. Appl. Math. Res. 3 (1) (2014) 71–80. [4] O. Hans, Random fixed point theorems, in: 1957 Transactions of the First Prague Conference on Information Theory, Statistical Decision Functions, Random Processses Held at Liblice Near Prague from November 28 to 30, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1956, pp. 105–125. [5] O. Hans, A. Spacek, Randon fixed point approximation by differentiable trajectories. Trans. 2nd Prague Conf. Information Theory, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1960, pp. 203–213. [6] G.A. Okeke, K.S. Eke, Convergence and almost sure T-stability for random Noortype iterative scheme, Int. J. Pure Appl. Math. 107 (1) (2016) 1–16. [7] C.H. Lee, W.J. Padgett, On random nonlinear contractions, Math. Syst. Theor. 11 (1977) 77–84. [8] G.S. Saluja, B.P. Tripathi, Some common random fixed point theorems for contractive type conditions in cone random metric spaces, Acta Univ. Sapientiae, Math. 8 (1) (2016) 174–190. [9] J.J. Nieto, A. Ouahab, R. Rodriguez-Lopez, Random fixed point theorems in partially ordered metric spaces, Fixed Point Theory Appl. 98 (2016) 2016. [10] M.C. Joshi, R.K. Bose, Some Topics in Nonlinear Functional Analysis, Wiley, New York, 1984. [11] W. Sintunavarat, Fixed point results in b-metric spaces approach to the existence of a solution for nonlinear integral equations, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 110 (2) (2016) 585–600.
mðvÞ aðt; vÞ bðt; vÞ jE aðt; vÞ Tðt; aðt; vÞÞ j þnðvÞ½ E þ bðt; vÞ Tðt; bðt; vÞÞ jE þqðvÞ½aðt; vÞ Tðt; bðt; vÞÞ jE þ bðt; vÞ Tðt; aðt; vÞÞ j a:s: E
Since kðvÞ < 1 a.s. Therefore Y(v) is a random nonlinear contraction operator on UðαÞ. By Theorem 3.1 and 3.2, there is a random variable y : ϕ → B which is the unique random solution of the operator (4). 5. Conclusion This research proved the existence and uniqueness of random fixed point for certain contractive mappings in complete partially ordered separable metric spaces. The existence and uniqueness of a random
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